Beyond CFD: Emerging methodologies for predictive simulation in cardiovascular health and disease

Abstract

Physics-based computational models of the cardiovascular system are increasingly used to simulate hemodynamics, tissue mechanics, and physiology in evolving healthy and diseased states. While predictive models using computational fluid dynamics (CFD) originated primarily for use in surgical planning, their application now extends well beyond this purpose. In this review, we describe an increasingly wide range of modeling applications aimed at uncovering fundamental mechanisms of disease progression and development, performing model-guided design, and generating testable hypotheses to drive targeted experiments. Increasingly, models are incorporating multiple physical processes spanning a wide range of time and length scales in the heart and vasculature. With these expanded capabilities, clinical adoption of patient-specific modeling in congenital and acquired cardiovascular disease is also increasing, impacting clinical care and treatment decisions in complex congenital heart disease, coronary artery disease, vascular surgery, pulmonary artery disease, and medical device design. In support of these efforts, we discuss recent advances in modeling methodology, which are most impactful when driven by clinical needs. We describe pivotal recent developments in image processing, fluid–structure interaction, modeling under uncertainty, and reduced order modeling to enable simulations in clinically relevant timeframes. In all these areas, we argue that traditional CFD alone is insufficient to tackle increasingly complex clinical and biological problems across scales and systems. Rather, CFD should be coupled with appropriate multiscale biological, physical, and physiological models needed to produce comprehensive, impactful models of mechanobiological systems and complex clinical scenarios. With this perspective, we finally outline open problems and future challenges in the field.

I. INTRODUCTION AND CLINICAL NEED

Cardiovascular modeling and simulation methods originated to provide clinicians and surgeons with predictive modeling capabilities to aid in optimizing and personalizing patient treatment plans. Likewise, it was intended that medical device design could be accelerated by testing proposed designs in a virtual setting before they are deployed in animal or clinical studies. This was initially accomplished by adapting existing physics-based predictive simulation technology for use in the biomedical space; such methods are routinely used by the automotive, aerospace, energy, and hi-tech industries to test and optimize designs before they are built. Early pioneers in this space developed idealized, and then patient-specific, models of blood flow and initial progress was driven by close partnerships between clinicians and engineers.^1–4 Adaptation of numerical technologies, such as finite element methods (FEMs), linear systems solvers, fluid–structure interaction (FSI) capabilities, and image segmentation algorithms, to biomedical problems led to rapid early growth in patient-specific modeling capabilities. However, the results of early attempts at predictive surgical planning based on “computational fluid dynamics (CFD) alone” were mixed. Indeed, this early experience laid bare the need to incorporate biological and physiological complexity that goes “beyond CFD.”

While the need for predictive treatment planning remains a primary motivation for simulation technology development, it is clearer than ever that models must account for mechanobiological response, physiologic adaptation, and patient heterogeneity across scales. Compared to traditional engineering applications, cardiovascular modeling brings a distinct set of challenges and complexities. These include anatomic structures beyond the limits of current image resolution, tissues that are mechanosensitive and adaptive, gene regulatory networks that differentially respond to mechanical stimuli, individual risk factors for disease initiation and progression, and autoregulatory systems that dynamically respond to changing conditions such as disease and exercise. Incorporating these effects across scales drives methodological developments unique to the cardiovascular setting. In addition, computational models are now proving crucial in basic scientific discovery beyond the original aims of treatment planning. As such, simulations are now being used to explain basic mechanisms of disease progression and cardiac development, settings in which mechanical forces are often instrumental but difficult to directly measure in vivo. With these deeper links between mechanical stimuli and mechanobiological response, exciting new possibilities for model-driven design are emerging.

In this review, we first briefly discuss the state of the art in patient-specific modeling capabilities, with a primary focus on core technology in image segmentation, patient-specific models, hemodynamics simulations, and fluid–structure interaction (FSI). We discuss the increasing clinical impact of this technology, with a particular focus on coronary and pulmonary applications, including congenital heart disease. Second, we describe important new modeling directions and technologies, including reduced order modeling (ROM), optimization, uncertainty quantification (UQ), multi-physics simulations, and whole heart modeling. Third, we review new and emerging capabilities at the interface of patient-specific modeling and biology, including growth and remodeling, thrombus formation, tissue engineering, and systems biology. We describe current and potential impacts of these methods to uncover fundamental mechanisms of disease progression and vascular and cardiac development. Finally, we discuss future challenges, including barriers to clinical adoption, and highlight emerging directions in tissue bioprinting and model-driven design. Before proceeding further, we briefly outline key open-source and open-data resources available to researchers in the field.

A. Open science resources

The field of cardiovascular biomechanics is fortunate to be replete with high-quality open science resources thanks to a strong culture of collaboration and sharing. Several open-source packages are available for image processing and segmentation including ITK-SNAP, 3D-Slicer, VMTK, and MITK.^5–8 In the area of modeling and simulation, a complete pipeline from anatomic model construction to finite element simulation and analysis is available from SimVascular⁹ and Crimson,¹⁰ which both include capabilities for FSI and boundary condition coupling to physiologic lumped parameter networks (LPNs). CircAdapt provides an open-source framework to simulate reduced order models of whole body circulation and includes the ability to simulate cardiovascular structural adaptation to experienced hemodynamic loads.¹¹ FEBio¹² provides rich capabilities in continuum mechanics, soft tissues, and, more recently, flow simulation. life^x similarly offers tools to simulate multiphysics, multiscale, and multidomain problems and is particularly optimized for the simulation cardiac mechanics problems.¹³ General purpose solvers may also be adopted for fluid mechanics and multi-physics applications in biomechanics, including OpenFoam¹⁴ and FEniCS.¹⁵ IBAMR (https://ibamr.github.io/) also provides high-quality flow and FSI solver technology using the immersed boundary (IB) approach. OpenCARP¹⁶ is a valuable resource for single-cell and multiscale cardiac electrophysiology simulations. Open-source packages for scientific visualization are also available and include ParaView¹⁷ and VisIt.¹⁸

Open data repositories have been developed to complement simulation packages and are increasingly important to support efforts in machine learning (ML) and artificial intelligence. The vascular model repository¹⁹ (https://www.vascularmodel.com) provides approximately one hundred fifty (and growing) complete projects containing medical images, anatomic models, and simulation files that are fully compatible with the SimVascular software (Fig. 1). The Cardiac Atlas Project²⁰ also contains multiple datasets containing cardiac images, segmented models, and accompanying tools for shape analysis, landmark detection, and motion tracking. In addition, several open-access resources exist for datasets of partial or whole-heart segmentations.^21–25 These repositories are of enormous value to the medical device community in light of recent efforts by the United States Food and Drug Administration (FDA) to increasingly incorporate modeling and simulation in the device approval process.

FIG. 1.

Subset of vascular models available in the vascular model repository.

A variety of commercial packages are also available for fluid and solid mechanics, finite element, and finite volume analysis, including Ansys Fluent, COMSOL, LS-DYNA, and Abaqus. A laudable effort in the field is the Living Heart Project led by Dassault Systémes, which has fostered a growing international ecosystem of industry, academic, and governmental collaborators to develop and validate highly accurate, personalized digital human heart models.²⁶ This group aims to establish foundational technology for cardiovascular in silico medicine in support of education and training, medical device design, testing, clinical diagnosis, and regulatory science.

II. PREDICTIVE SIMULATION METHODS

A. Anatomic model construction and image processing

Cardiovascular disease patients exhibit heterogeneity in anatomy, material properties, physiology, and genetics. Patient-specific, as opposed to idealized or population averaged, models are, thus, needed to accurately capture variable anatomy and hemodynamic conditions. The patient-specific modeling pipeline typically begins with image processing and segmentation. This is an often laborious process requiring “by hand” user segmentation and expertise in medical imaging to create accurate models. A primary method for anatomic model creation is the “path planning” approach in which centerline vessel paths are created, the vessel lumen is segmented in 2D contours normal to the paths, and the segmentations are “lofted” to create a complete 3D anatomic model. This method forms the backbone of the modeling pipeline in the SimVascular and Crimson packages and has been employed in numerous application-oriented studies.^27–32 Alternative 3D segmentation approaches are also popular for model construction and are available in packages such as ITK-Snap and VMTK. Common methods include colliding fronts, region growing techniques, and active contours. Extensive reviews of available choices are found elsewhere.³³

Automating the image segmentation process is crucial to enable clinical trials and studies with large patient cohorts. Currently, the substantial time required for model construction poses a major barrier to both clinical adoption and the use of modeling in large studies. While traditional approaches such as level set and thresholding methods offer some automation capabilities, these methods often require extensive parameter tuning, which largely negates any gains in efficiency.

Machine learning (ML) and convolutional neural networks (CNNs) have, thus, recently been adopted for both vascular and cardiac image segmentation with promising results. Vessel lumen regression using neural networks dramatically reduced user time for model construction in a recent study and demonstrated good accuracy, particularly in larger vessels.^34,35 Uncertainties using this approach were quantified and found to be on par with user-to-user variability in by-hand image segmentation. Automated segmentation of coronary angiography images was successfully performed using the AngioNet platform with CNNs in recent work.³⁶ Cardiac segmentation with ML, including all four heart chambers and major vessels, is also showing promise, and excellent results were recently achieved using graph neural networks with a mesh decoding module.^25,37,38 ML has also been used for the automatic assessment of image quality, providing an operator-independent method of standardizing datasets for further processing, including automatic segmentation.³⁹

B. Hemodynamics simulations and fluid–structure interaction methods

Cardiovascular modeling requires handling multiple coupled physical processes, including the interaction of fluids, solids, electrical propagation, active contraction, and transport. While simulations of flow and fluid–structure interaction are relatively well established, multi-physics models are an active area of research that will be discussed in Sec. IV B.

Unstructured mesh generation in complex anatomic models is typically performed with consideration of boundary layers, adaptation to velocity gradients, and areas of local refinement.⁴⁰ Hemodynamics simulations are then carried out using computational fluid dynamics (CFD), which can be performed with a variety of numerical methods, including finite element methods (FEM) and finite volume techniques. Methods for rigid wall CFD in patient-specific models are generally well established, particularly with simple outflow boundary conditions such as resistance and Windkessel models.^41,42 Even with simple boundary conditions, it is important to replicate physiologic conditions and tune parameter values to match available clinical data. Indeed, simulations can produce dramatically non-physiologic results when naive (e.g., zero pressure) boundary conditions are applied.⁴¹ More extensive physiologic models are also now routinely applied using 3D–0D approaches, in which the 3D domain is numerically coupled to an open or closed-loop lumped-parameter network (LPN) of the circulation, solved numerically.^43–46 LPN models make a hydraulic analogy with electrical circuits and can include heart models, nonlinear elements, and coronary physiology. This coupling also necessitates special considerations in the linear system solution due to ill-conditioning, and custom preconditioning and bi-partitioned linear solver strategies have been developed.^47,48 Backflow stabilization is often needed to avoid numerical instabilities caused by flow reversal at the model outlets.⁴⁹ 3D simulations have been similarly coupled to 1D models to capture wave propagation in compliant models.^50,51 Reduced order modeling strategies are discussed further in Sec. IV A.

FSI is critical to capture wall deformation, wave propagation, and tissue mechanics in cardiovascular models and is increasingly well established. The arbitrary Lagrangian–Eulerian (ALE) method has been adopted for cardiovascular FSI with thick-walled vessels and variable material properties in several studies, including application to complex geometries such as cerebral aneurysms and single ventricle models.^52,53 In the ALE framework, fluid–structure behavior is governed by

$$0={\frac{\partial \rho }{\partial t}|}_{\chi }+(\mathbf{v}-\widehat{\mathbf{v}})·{\nabla }_x\rho +\rho {\nabla }_x·\mathbf{v},$$ (1)

$$\mathbf{0}=\rho {\frac{\partial \mathbf{v}}{\partial t}|}_{\chi }+\rho (\mathbf{v}-\widehat{\mathbf{v}})·{\nabla }_x\mathbf{v}-\rho \mathbf{b}-{\nabla }_x·\mathit{\sigma }$$ (2)

in the Eulerian fluid domain, where ρ is the density, χ represents the spatial coordinate with respect to a suitable reference frame moving with the domain velocity $\widehat{\mathbf{v}}$, v is the velocity, $\mathit{\sigma }$ is the fluid stress tensor, and b is the body force per unit mass. The Lagrangian solid domain is governed by

$$\mathbf{0}=\text{div}{\mathit{\sigma }}^s+{\mathbf{b}}^s-{\rho }^s{\mathbf{a}}^s,$$ (3)

where ${\mathit{\sigma }}^s$ is the Cauchy stress in the solid domain, ${\mathbf{b}}^s$ is the body force per unit mass in the solid domain, ρ^s is the density in the solid domain, and ${\mathbf{a}}^s$ is the acceleration in the solid domain. The tractions, displacements, and velocities are constrained to match at the fluid–structure interface.

Figure 2 shows a clinical application of FSI to aortic dissection.²⁷ This example incorporates vessel wall prestress and external tissue support, which has proven to be crucial for damping unphysical oscillations and avoiding unrealistic over-distension of pressurized models.^54–56 While most FSI studies to date have used linear material models, sophisticated nonlinear models of vascular tissues based on sound experimental data are widely available in the continuum mechanics community and should be adopted to increase the realism of FSI simulations.^57,58 Efficiency gains have been made with the coupled momentum method (CMM), introduced as an alternative to ALE for small deformation vascular FSI,⁵⁹ and this method has been validated against in vitro experiments and widely adopted.^60–64

FIG. 2.

FSI model of aortic dissection with vessel wall prestress and external tissue support. Streamlines (left) in systole (top) and diastole bottom and pressure (right). True and false lumen are clearly visible with streamlines entering the false lumen at the focal entry tear. Adapted from Ref. 27.

Key advances in FSI methodologies have been introduced recently, overcoming prior limitations of both ALE and CMM methods for vascular simulations. ALE methods traditionally employed dichotomous formulations for the fluid and solid subproblems, creating challenges and disparities in numerical schemes for modeling the coupled system. While the constitutive relations for finite elasticity were traditionally formulated in terms of the Helmholtz free energy (or strain energy), this ceases to be a proper potential in the incompressible limit, leading to ill-conditioned matrix systems. To overcome this issue, recent work of Liu et al. employed a Legendre transformation, allowing the use of Gibbs free energy to describe material behavior in both the incompressible and compressible regimes.⁶⁵ In this setting, the following unified ALE description emerges, governing the conservation of mass and momentum in the fluid and the solid,

$$0={\beta }_{\theta }{\frac{\partial p}{\partial t}|}_{\chi }+{\beta }_{\theta }(\mathbf{v}-\widehat{\mathbf{v}})·{\nabla }_{x}p+{\nabla }_x·\mathbf{v},$$ (4)

$$\mathbf{0}=\rho {\frac{\partial \mathbf{v}}{\partial t}|}_{\chi }+\rho (\mathbf{v}-\widehat{\mathbf{v}})·{\nabla }_x\mathbf{v}-{\nabla }_x·{\sigma }_{\text{dev}}+{\nabla }_{x}p-\rho \mathbf{b},$$ (5)

where ρ is the density, p is the pressure, ${\beta }_{\theta }:=(\partial \rho /\partial p)/\rho$ is the isothermal compressibility coefficient, χ represents the spatial coordinate with respect to a suitable reference frame moving with the domain velocity $\widehat{\mathbf{v}}$, x is the spatial coordinate in the Eulerian domain, v is the velocity, ${\sigma }_{\text{dev}}$ is the deviatoric component of the Cauchy stress, and b is the body force per unit mass. Constitutive modeling determines the analytic forms for $\rho (p),{\beta }_{\theta },$ and ${\sigma }_{\text{dev}}$ to close the system. This system governs both viscous fluids and elastic solids and thus bridges the gap between methods for CFD and structural dynamics. This simplifies FSI analysis by producing a unified formulation requiring a single numerical solution strategy.

The above formulation spurred important further developments. Discretizing the nonlinear partial differential equation (PDE) system in Eqs. (4) and (5) in conjunction with the Newton–Raphson method produces a large linear system of sparse matrices, the solution of which constitutes the bulk of the computational time for simulation. A robust and efficient iterative method with nested block preconditioning was introduced to leverage the structure of the system for efficiency gains.^66,67 Within the nonlinear solution procedure, a block factorization is performed for the consistent tangent matrix to decouple the kinematics from the balance laws. Within the linear solution procedure, another block factorization is performed to decouple the mass balance equation from the linear momentum balance equation. A nested block preconditioner was introduced to combine the Schur complement reduction approach with the fully coupled approach, yielding performance gains compared to standard methods in several full-scale cardiovascular examples. Further developments led to a formulation for handling viscoelasticity in finite deformation linear models.⁶⁸ Improvements in the temporal accuracy for the pressure were also achieved via modifications to the generalized-α time advancement method.⁶⁹

A novel reduced unified continuum (RUC) formulation was then developed for efficient vascular FSI building on the above technologies.⁷⁰ In contrast to the CMM method, the underlying derivation does not require the use of a fictitious body force in the elastodynamics sub-problem, resulting in a direct reduction of the ALE method. The generalized-α method was adopted for the monolithic problem, avoiding prior issues of high-frequency oscillations. This method was formulated with a three-level nested block preconditioner which demonstrated robust performance over a wide range of physical parameters. Verification against Womersley's deformable wall theory showed excellent agreement.

The immersed boundary (IB) method, first introduced by Peskin for ventricular flow simulations,⁷¹ has also been widely adopted for FSI, particularly for cardiac geometries and heart valve simulations with large leaflet deformation and contact. The original formulation describes the momentum, viscosity, and incompressibility of the fluid–solid system in the Eulerian form and the deformations, stresses, and resultant forces of the immersed structure in Lagrangian variables. The Eulerian and Lagrangian frames are then linked using integral transforms with Dirac delta function kernels. Recent noteworthy developments in IB methodology include sharp interface and discrete immersed surface methods.^72–74 Handling of incompressible solids has been improved in recent work introducing stabilization approaches for hyperelastic immersed surfaces.⁷⁵ Experimental validation of these methods in artificial heart valves was successfully performed using in vitro pulse duplicators.⁷⁶ Design-based models of the aortic and mitral valves provide an elegant means to obtain realistic valve models that adhere to first principles.⁷⁷ Such methods have been used recently to study mitral valve hemodynamics and the effects of trabeculae in the left ventricle.^78,79 Technologies and treatment strategies, including experimental and simulation methods, for aortic and mitral valve repair are further discussed in recent reviews.^80–82

Many of the FSI methods described above are available in the open-source and commercial simulation suites described in Sec. I A. SimVascular supplies tools for simulating both ALE and CMM cardiovascular simulations and is also capable of assigning user-defined LPN boundary conditions. Crimson is able to simulate CMM FSI with LPN boundary conditions with ALE methods currently under development. The open-source solvers FEBio, OpenFoam, and FEniCS are also capable of solving FSI problems using implementations of the ALE methods but are less optimized for applying cardiovascular LPN boundary conditions.^83,84 Open-source implementation of IB methods are available through IBAMR.

III. CLINICAL IMPACT IN ADULTS AND CHILDREN

Computational models of hemodynamics have been broadly applied to a wide range of adult and pediatric diseases spanning cerebral and abdominal aortic aneurysms, vascular surgery and endovascular devices, stroke and embolism, the thoracic aorta, and others.^86–89 Models have also been instrumental in accelerating the medical device design pipeline, allowing for inexpensive design optimization prior to full-scale animal and human studies.^90–93 Here, we focus primarily on two areas of clinical need in cardiovascular disease: coronary artery disease and pulmonary vascular disease. These examples illustrate the clinical impact and further motivate the need to move beyond CFD to multiscale methods incorporating mechanobiology.

A. Coronary artery hemodynamics

Perhaps the most widespread translational impact of CFD methods to date has been in the diagnostic assessment of coronary artery disease severity. Patients with ischemic vascular disease caused by atherosclerosis often undergo invasive cardiac catheterization procedures to measure fractional flow reserve (FFR), a measure of pressure drop across stenosis under hyperemic conditions. The use of FFR to guide interventional decisions (angioplasty, stenting) has been shown to significantly reduce death, myocardial infarction, and repeat revascularization in large randomized clinical trials.^94,95 Technology to noninvasively assess FFR based only on computed tomography (CT) imaging, ${\text{FFR}}_{\text{CT}}$, was introduced by HeartFlow, Inc., and granted FDA approval in 2014. High-fidelity anatomic models are constructed from patient CT imaging data, and steady flow CFD simulations are performed with specialized coronary physiologic boundary conditions.^61,96 Large-scale clinical trials have demonstrated superior sensitivity and specificity in diagnostic performance using computational predictions of ${\text{FFR}}_{\text{CT}}$ compared to traditional imaging methods.⁹⁷ Further studies in clinical efficacy showed that two-thirds of patients resulted with a different management plan when using noninvasive CFD-based analysis and a substantial reduction in unnecessary invasive and expensive catheterization laboratory procedures.^98,99 This technology is currently being extended to interventional planning applications to aid clinicians in decisions on angioplasty and stent placement, particularly in patients with multiple lesions. Simulations also hold promise in surgical planning for coronary artery bypass graft (CABG) patients, and have been applied to compare competing graft configurations, including sequential, multiple and Y-shaped graft designs.¹⁰⁰ Optimization has also been used to identify graft designs that minimize areas of low wall shear stress (WSS) and atherosclerotic plaque deposition.^101,102

Patients with multi-vessel, diffuse coronary artery disease, for whom angioplasty and stenting are not recommended, typically undergo CABG surgery. Limited availability of arterial grafts necessitates the use of saphenous vein grafts (SVGs) in 95% of CABG patients.^103,104 However, long-term benefits of CABG are severely limited by obstructive SVG disease, which is caused by thrombosis (early), intimal hyperplasia (intermediate), or atherosclerosis (long-term).¹⁰³ Complete SVG obstruction occurs in 40%–50% of grafts within 10 yr,^105,106 and recent data indicate that 25% of SVGs have severe stenosis or occlusion within 12–18 months.¹⁰⁷ SVG failure is associated with significantly worse long-term clinical outcomes.^108,109 SVG adaptation is driven by changes in mechanical stimuli that are sensed by vascular cells.^110–114 CABG surgery transfers the SVGs from the low-pressure venous system to the high-pressure, high-flow arterial system (increasing pressure 20-fold and WSS fourfold), which is followed by vein arterialization.^115,116 This abrupt and sustained increase in hemodynamic loads induces cellular proliferation and apoptosis, phenotypic changes, and inflammatory response, each of which are thought to contribute to vein graft maladaptation. Thus, modulating hemodynamic-induced immuno-mechano-biological responses is likely key to ensuring long-term SVG patency.^117–119

Blood flow simulations have confirmed significant differences in WSS and strain when comparing arterial and venous grafts¹²⁰ and have revealed correlations between WSS and SVG stenosis after CABG.¹²¹ To model SVG growth and remodeling, constrained mixture theory has been adopted, which accounts for different rates of production and removal of extracellular matrix constituents and cells within the wall.^117,122 Simulations revealed that gradual, rather than abrupt, loading of SVGs following CABG might ameliorate failure risk.¹²² Implementation of this concept has been explored via the use of an external sheath to mechanically support vein grafts subjected to arterial pressure. Studies in animal models confirmed that external scaffolds significantly reduce neointimal hyperplasia and medial wall thickening in vein grafts after arterial bypass and can prevent vein graft dilatation.^123,124 The randomized, self-controlled VEST trial of 30 human subjects undergoing multi-vessel CABG demonstrated significantly reduced neointimal hyperplasia in SVGs supported by a cobalt-chromium external stent compared to unsupported SVGs, although overall patency did not differ significantly at one year after surgery.¹²⁵ Recent modeling work indicates promise of a degradable external sheath,¹²⁶ which may overcome issues of inflammation observed in prior studies.

Advances in imaging methods are now enabling increasingly detailed assessment of coronary perfusion using computed tomography (CT). These measures have been demonstrated to accurately identify coronary artery disease when used in conjunction with FFR acquired with computed tomography angiography¹²⁷ and show promise for increasing the accuracy of coronary CFD simulations by generating coronary tree boundary conditions based on measured perfusion zones.¹²⁸

Simulations of coronary hemodynamics also show promise for treatment planning and risk stratification in pediatric cardiology. In Kawasaki disease, hemodynamic simulations have been used to improve thrombotic risk assessment for patients with coronary artery aneurysms; see, e.g., Fig. 3 (top). In these studies, high residence times and large areas of WSS in aneurysms were shown to be better indicators of thrombotic risk than purely geometric methods of risk stratification.³⁰ In a recent computational study featuring the patient shown in Fig. 3, pressure drops across aneurysms have been shown not to be dramatic over a cohort of 15 patients.¹²⁹ However, the presence of aneurysms was associated with rapid pressure drops in the distal parts of the coronary tree, which indicates an increased risk of myocardial ischemia. FSI simulations have also been applied to improve the understanding of hemodynamics in patients with anomalous origin of the coronary arteries.¹³⁰

FIG. 3.

Examples of clinical applications. Top: Numerical approximation of FFR in coronary arteries with aneurysms caused by Kawasaki disease: streamlines of velocity field (left) and pressure distribution (averaged over time) in hyperemia (right). The aneurysms are marked with boxes on the right. Bottom: Y-graft Fontan model.⁸⁵ During inspiration, the inferior vena cava (IVC) flow is bifurcated by the Y-graft and channeled to the pulmonary arteries. During expiration, retrograde flow in the IVC allows the superior vena cava (SVC) flow to enter the right limb of the Y-graft.

B. Pulmonary arteries and single ventricle physiology

Pulmonary vascular circulation is supplied by the right ventricle and is responsible for blood oxygenation, which occurs where the alveoli interface with the bronchial and lymphatic systems. It has a complex branched structure, with the main pulmonary artery arising from the right ventricular outflow tract, branching into left and right main pulmonary arteries, and subsequently into two lobar branches each and then segmental and sub-segmental branches. This structure generally parallels the bronchial branches with names corresponding to the associated bronchopulmonary segments. Typically, the extensive branching of the pulmonary tree allows for even distribution of blood to the alveoli interface. Pulmonary diseases, including pulmonary hypertension (PH) and congenital heart defects, that disrupt the natural flow of blood through the pulmonary arteries can lead to altered gene expression, unbalanced perfusion to the lungs, deoxygenation, adverse cardiovascular remodeling, and increased right ventricular workload. Computational modeling of blood flow has provided new insights into how pulmonary diseases disrupt normal pulmonary and cardiac function.

In pulmonary hypertension (PH), adverse growth and remodeling as well as increased ventricular workload can lead to eventual cardiac failure. However, the hemodynamics that give rise to these conditions are difficult to measure in vivo.^131,132 Quantification of pulmonary artery hemodynamics using 4D MRI has shown decreased WSS, vorticity, and helicity in the main pulmonary artery of patients with PH, but the resolution limitations of 4D MRI make it unsuitable for quantifying hemodynamics in the smaller, distal branches where disease is thought to initiate.^133–135 Computational simulations of the pulmonary tree can allow for detailed quantification of hemodynamics in a greater portion of the pulmonary arterial tree than is possible with current imaging techniques. Blood flow simulations showed that severe PH is associated with decreased WSS in the proximal arterial tree and increased WSS in the distal pulmonary tree, providing evidence that PH can trigger maladaptive mechanobiological responses along the entirety of the pulmonary artery tree. In addition, the oscillatory shear stress was found to be lower in the main pulmonary artery in patients with severe PH.^28,136,137

It is important to carefully consider the boundary conditions for pulmonary artery simulations. While many studies have employed traction-free or constant resistance outlet conditions, distal pulmonary artery impedance significantly impacts upstream hemodynamics and is a significant predictor of pulmonary hypertension severity.^138,139 To address this, the impedance of the distal pulmonary vasculature has been modeled using morphometric trees built using pulmonary scaling laws. Morphometric trees have been able to model pulmonary hypertension stemming from both arterial rarification and stiffening, showing their ability to capture in vivo trends.¹⁴⁰ Future studies examining mechanisms of PH disease progression must account for both mechanical stimuli and biologic response at multiple scales of the pulmonary artery (PA) tree, given the disparate effects of PH in the proximal and distal tree.

Pulmonary stenosis, or narrowing of the pulmonary vessel lumen, is another significant cause of pulmonary flow disruption. Pulmonary stenosis is commonly associated with congenital heart diseases, including Williams syndrome, Alagille syndrome, and tetralogy of Fallot. In many cases, multiple stenotic lesions are present and distributed across the main and peripheral pulmonary artery branches. While correcting major lesions with a combination of stenting and balloon angioplasty has the ability to reduce pressure gradients across individual lesions, this may only lead to minor improvements in right ventricular pressures in a significant portion of patients.^141–143 Computational models of the pulmonary arterial tree have shown that stenting only proximal lesions is often insufficient to reduce overall vascular resistance and thus right ventricular pressures.¹⁴⁴ In these cases, total vascular resistance often remains high due to the remaining peripheral pulmonary lesions and distal vasculature. Detailed simulations of treatment planning scenarios may aid clinicians in identifying the most critical targets for intervention. Modeling the distal pulmonary tree using reduced order methods has also shown that pulmonary stenosis may lead to adverse downstream remodeling that may cause persistent vascular resistance increase even after universal resolution of pulmonary stenoses.¹⁴⁵ Incorporating vascular growth and remodeling will be critical to predicting long-term PA adaptation and disease progression.

Computational modeling has also played a significant role in investigating palliation strategies for single ventricle physiology involving the pulmonary vasculature. While single ventricle physiology has numerous presentations, it is typically treated through three-stage surgical palliation that results in the Fontan physiology.¹⁴⁶ Because single ventricle physiology affects the entire circulation, closed-loop LPNs have been an important tool in simulating the effect of modifying cavopulmonary connections in single ventricle patients (Fig. 4). Early computational studies of the Norwood operation, a version of the first stage of palliation, used LPN models of single ventricle circulation to quantify the effect of varying the arterial-pulmonary shunt size. This work showed that while larger shunts may lead to increased pulmonary flow, they can result in lower systemic oxygen delivery.^147,148 The multiscale modeling approach was then employed to investigate the second stage of single ventricle palliation, the Glenn procedure, and LPNs were coupled directly to 3D models of patient-specific geometries in multiscale 0D–3D simulations.¹⁴⁹ This allowed for the direct investigation of local hemodynamics in the cavopulmonary system and the modification of LPN parameters to model closed-loop exercise states. Closed-loop multiscale simulations have been experimentally validated against in vitro models that utilize pumps and patient-specific, 3D-printed models, with excellent results.^150–152 This body of work showed that patient-specific geometries and boundary conditions could greatly impact the modeled performance of the single ventricle palliation stages and emphasized the need for high-fidelity simulations for use in surgical planning and optimization.

FIG. 4.

An example of multiscale modeling in a Fontan patient which allows for quantification of local hemodynamics and estimation of whole-body behavior in response to cavopulmonary changes. Adapted from Refs. 159 and 161.

While Fontan palliation has led to increased survival of single ventricle patients, Fontan physiology also causes elevated central venous pressure, reduced cardiac preload, and reduced pulmonary pulsatility, which bring associated morbidities such as liver fibrosis, protein losing enteropathy, arrythmia, exercise intolerance, and adverse cardiac and vascular remodeling. Because these morbidities are associated with the hemodynamic inefficiencies and elevated venous pressures of the Fontan pathway, energetics of the Fontan connection have been carefully studied as the basis for surgical optimization. At rest, simulations have shown that differences in Fontan energetics lead to a minimal difference in global circulatory physiology, including pressure–volume loops and cardiac output.¹⁵³ However, differences are accentuated during exercise. Simulations have shown that increased total cavopulmonary resistance significantly decreases the ability of the Fontan circulation to increase cardiac output during exercise, and further studies showed that energy loss increases nonlinearly in the total cavopulmonary connection (TCPC) during exercise even when normalized to head loss or resistance, demonstrating the limited ability of the TCPC to accommodate exercise conditions^154,155 Experiments have been performed to identify primary causes of energy loss and hemodynamic inefficiencies in the Fontan pathway. While energetics were found to be similar across the lateral tunnel and extracardiac variations of the TCPC, simulations showed that in extracardiac Fontan conduits, head-on collision of flow results in increased energy loss.^2,156,157 This led to the development of an offset conduit design that prevented head-on flow collision. However, computational models of the offset conduit design showed uneven hepatic flow distribution to the pulmonary arteries. To address this issue, the bifurcated Y-graft, shown in Fig. 3 (bottom), was introduced. This improved energy loss, SVC pressure, and hepatic flow distribution compared to the offset design in computational studies that were later validated clinically.^85,158 As their predictive power has grown, computational simulations continue to be used in investigations of new palliative techniques, including surgical alternatives to the first stage of palliation, such as the assisted bidirectional Glenn and models of tissue-engineered vascular grafts in the Fontan pathway.^159,160

IV. EMERGING METHODOLOGIES AND APPLICATIONS

A. Reduced order modeling and uncertainty quantification

The CFD techniques discussed throughout this paper allow us to describe cardiovascular systems with an exceptional level of detail. However, this degree of accuracy is often associated with computational costs and resources incompatible with many practical scenarios. Reduced order models (ROMs) are computationally cheap alternatives to high-fidelity methods such as the finite element or finite volume methods. While high-fidelity methods often require hours or days to perform a single simulation on supercomputers, ROMs are typically associated with running times of the order of seconds or minutes and can be executed on inexpensive hardware (e.g., laptops). The dramatic increase in performance is linked to a loss in accuracy and detail, but these often remain sufficiently high for many clinical applications.

The most common ROMs in cardiovascular applications are 0D and 1D models. The former are sometimes called lumped-parameter networks or LPNs. In 0D formulations, regions of the cardiovascular system are modeled as electric circuits where the pressure drop across any two points of the model $\Delta P$ and the flow rate Q play the role of voltage and electric current, respectively. The laws governing pressure drop $\Delta P$ and flow rate Q are¹⁶²

$$\Delta P=RQ,\hspace{1em}Q=C\Delta \dot{P},\hspace{1em}\Delta P=L\dot{Q},$$ (6)

where R, C, and L are quantities associated with elements commonly found in electric circuits, namely, resistors, capacitors, and inductors, respectively, and $\dot{P}$ and $\dot{Q}$ are the time derivatives of pressure and flow rate. These quantities can be expressed in terms of the geometrical and structural properties of the vessels under the assumption of a fully developed parabolic velocity profile.^163,164 LPNs are often used in conjunction with 3D models to describe the effects of the downstream vasculature, for instance, using resistor-capacitor-resistor (known as RCR or Windkessel) boundary conditions.⁴² Moreover, 0D models have been used to model the heart, systemic, pulmonary, and coronary circulations to study congenital heart disease^149,165 and coronary artery disease.^166–168 The performance of 0D and 3D models has been compared in several studies, for example to estimate flow rate in cerebral artery aneurysms,¹⁶⁹ fractional flow reserve (FFR) in coronary arteries,¹⁷⁰ pressure drop across branch pulmonary artery stenosis,¹⁷¹ and pressure and flow rate distributions in a variety of different cardiovascular regions.¹⁷²

In contrast to LPNs, some geometrical information from the cardiovascular model of interest is retained in 1D formulations. Here, the branches in the vessel of interest are approximated as curves (usually, the branch centerline) in 3D space and flow dynamics are described in terms of pressure and flow rate along these curves. Under the assumption of a parabolic flow profile, the 1D equations read as

$$\frac{\partial A}{\partial t}+\frac{\partial q}{\partial z}=0,$$ (7)

$$\frac{\partial q}{\partial t}+\frac{\partial }{\partial z}\left(\frac{4}{3}\frac{q^2}{A}\right)=-8\pi \nu \frac{q}{A}+\nu \frac{{\partial }^{2}q}{\partial z^2}-\frac{A}{\rho }\frac{\partial p}{\partial z},$$ (8)

where $p=p(z,t),\hspace{0.17em}q=q(z,t)$ and $A=A(z,t)$ are pressure and flow rate of blood and the area of the vessel lumen at axial location z, respectively. The kinematic viscosity ν is typically set to $\nu =3.77\times {10}^{-2}\hspace{0.17em}\mathrm{s}\hspace{0.17em}{\text{cm}}^{-2}$. System (7) and (8) needs to be complemented with a constitutive law to model the structural response of the vessel to blood flow. A popular choice is the Olufsen material model which builds on simple reduced-order linear elasticity by using an exponential relationship based on empirically fitted data.¹⁷³ Reduced order 1D models can be coupled with 3D models of regions of the cardiovascular tree where higher resolution is required.^51,174,175 For example, 1D models have been coupled to 3D models of regions of interest to study coronary hemodynamics.¹⁶⁸ Due to their computational efficiency, 1D models can also be used to describe blood dynamics in all the major vessels of the cardiovascular tree.^176–178 These ROMs have also been successfully employed to compute initial conditions for 3D models with the goal of reducing the number of cardiac cycles needed to reach a periodic state.¹⁷⁹ Comparisons between 1D and 3D models have focused on the approximation of aortic flow,^180–182 FFR in coronary arteries,^183–185 and blood dynamics in the Circle of Willis^186,187 and in the whole arterial tree.^188–192 A recent study¹⁶⁴ also found good agreement between 0D/1D and 3D results for 72 geometries taken from the Vascular Model Repository.¹⁹ Finally, some works have compared 1D results with experimental data in large networks of arteries and veins^176,188,193 and in the global arterial tree.¹⁹⁴

Other popular ROMs are data-driven, i.e., they aim to exploit past simulation results to perform future simulations more efficiently. In proper-orthogonal decomposition (POD) and reduced basis (RB) methods,^195,196 for example, a small number of pressure and velocity modes is generated through singular value decomposition (SVD) from a database of high-fidelity solutions. The pressure and velocity fields are then found as linear combinations of these basis functions and, as a result, the size of the linear systems to be solved in each time step is considerably smaller than in the high-fidelity counterpart. Notable applications of the RB method to cardiovascular flow are, for example, the study of parameterized coronary artery bypass grafts and shape optimization,^197–200 the solution of inverse problems to retrieve vessel structural properties,¹⁹⁹ and the investigation of the effects of stenosis severity on flow.²⁰¹ It is worth noting that other approaches not based on data-driven ROMs have been used for shape optimization in cardiovascular problems; e.g., the Surrogate Management Framework (SMF) has been successfully used to optimize the assisted bidirectional Glenn procedure¹⁶⁰ or a Y-shaped baffle for the Fontan surgery.²⁰² To increase the applicability POD and RB techniques to geometries not considered during the reduced basis generation, some studies have focused on decomposing vessels into multiple geometrical building blocks, each equipped with a small number of basis functions.^203–205 Another promising class of data-driven techniques that has gained attention in the cardiovascular research community in recent years is physics-informed neural networks (PINNs).²⁰⁶ These can be trained on 1D data to derive surrogate 1D models²⁰⁷ or on the Navier–Stokes equations to approximate wall-shear stress from a set of sparse measurements of blood velocity and incomplete boundary conditions.²⁰⁸

ROMs are especially useful in many-query scenarios, namely, in cases where cardiovascular models need to be run many times for different values of parameters associated with the properties of the system. A prominent area of research requiring a great number of simulations is uncertainty quantification (UQ). Cardiovascular simulations are associated with an intrinsic level of uncertainty stemming from the challenges in setting appropriate boundary conditions, variability in material properties of vascular tissue, and uncertain vascular geometry due to operator-dependent image segmentation processes. In high-stakes situations, therefore, estimating quantities of interest via a single deterministic model evaluation is not advisable, and assigning confidence intervals and probability distributions to such quantities is of paramount importance. In UQ, this is achieved by sampling uncertain model parameters from probability distributions informed by patient data or by the literature. Specifically, in Monte Carlo (MC) approaches, the output of interest—which is generally a function of the pressure and velocity fields obtained from (expensive) 3D simulations—is evaluated for many different values of the uncertain parameters. Then, statistics on the quantities of interest are computed. This approach led to the study of the impact of geometric uncertainty on patient-specific blood flow simulations using convolutional neural networks with dropout layers for automatic segmentation.³⁵ Similarly, sensitivity with respect to the segmentation of coronary artery vessel lumen has been estimated using anisotropic kernel regression.²⁰⁹ Multi-level Monte Carlo (MLMC), multi-fidelity Monte Carlo (MFMC), and multi-level multi fidelity (MLMF) estimators have been devised to alleviate the computational burden of standard MC approaches by relying on a small number of expensive high-fidelity solutions and a large number of low-fidelity ones obtained using coarse meshes, 0D and 1D ROMs, and combinations of the two, respectively.^210,211 Recent efforts have also considered other UQ approaches such as Markov chain Monte Carlo (MCMC), stochastic collocation, and multi-wavelet stochastic expansion to investigate the effects of a variety of uncertain parameters on coronary flow.^212,213 Another area in which a large number of cardiovascular model evaluations is required and in which, therefore, the use of ROMs becomes particularly effective, is boundary condition tuning, such as using Bayesian frameworks based on adaptive MCMC sampling.²¹⁴

B. Multi-physics simulations and whole heart modeling

While the field of vascular flow simulations is relatively mature, complex clinical applications (e.g., complex congenital heart disease, cardiomyopathy, pulmonary hypertension) increasingly require the use of cardiac or coupled vascular-cardiac models. Such models bring added complexity due to their multi-physics nature.²¹⁵ From a physical perspective, modeling cardiac function is a coupled problem of electrophysiology, solid mechanics, and fluid dynamics.^216,217 However, the mathematical description and numerical solution of fully coupled cardiac models remains highly challenging. Current research efforts are, therefore, geared toward speeding up computations of highly detailed models²¹⁸ and developing cheaper low-order models that retain physical accuracy.^219–221 Depending on the clinical application, the three physical fields (electrical, solid, fluid mechanics) of the cardiac problem are coupled with varying levels of model fidelities. From a computational viewpoint, the coupling of multi-physics heart simulations can be roughly categorized by electromechanical and fluid–structure interactions, which are described below.

The electrophysiology problem solves for the transmembrane potential and is usually modeled by the monodomain or bidomain equations. For an overview of cell- and organ-scale electrophysiological models, we refer to the review of Trayanova and Rice.²²² The active contraction of the cardiomyocytes has been modeled both with active stress and active strain models, whose differences are discussed by Ambrosi and Pezzuto.²²³ Different approaches exist to couple the electrical activation of the heart with cardiomyocyte contraction. In the weakly coupled approach,^224,225 the electrical wave propagation is solved first on the undeformed cardiac geometry, typically with a finer mesh and a smaller time step than the mechanical problem, and then used as a boundary condition to calculate active mechanical contraction. The strongly coupled approach simultaneously solves the electromechanical problem, including feedback of the deformation onto the electric current flow.²²⁶ While weak coupling ignores this feedback, its approach typically offers a simpler implementation and reduced computational effort. A popular benchmark for different discretization methods in cardiac electrophysiology was proposed by Niederer et al.,²²⁷ comparing activation wavefronts in a cuboid domain. An overview of applications of whole-heart modeling to cardiac electrophysiology is given by Trayanova.²²⁸

Two popular approaches for the cardiac fluid–structure interaction are 3D–0D and 3D–3D coupling. The 3D–0D approach couples detailed models of 3D solid mechanics with 0D reduced models of fluid dynamics, modeling hemodynamics either with open-loop²²⁹ or closed-loop^230,231 LPNs. 3D–0D coupled models are popular in applications focusing on myocardial mechanics, such as predicting cardiac growth and remodeling in hypertrophy with detailed microstructural solid models.²³²

On the other hand, studies on the influence of intracardiac blood flow on disease initiation and progression warrant a finely resolved 3D model of cardiac fluid dynamics. An excellent overview of coupling fluid-mechanics-electrophysiology was recently published by Verzicco.²³³ An important example of a clinical application that requires detailed models of fluid dynamics is valvular heart disease. Heart valves are thin structures undergoing large displacements while changing their status from closed to open, which makes modeling their interaction with blood flow challenging.²³⁴ Here, a popular computational method is the immersed boundary method^71,235,236 which has been applied to left-heart²³⁷ and four-chamber heart models, including valves.²³⁸ In a simplified approach, the valve is implicitly represented by a resistive surface that represents the valve status, open or closed, with a 0D valve model.^239,240 Another clinical application that demands resolving detailed 3D intracardiac fluid dynamics is determining the thrombogenic risk in the left atrial appendage during atrial fibrillation and the associated stroke risk.²⁴¹ Here, fluid dynamics inside the left atrium can identify regions where blood flow cannot wash out the chamber sufficiently.^242,243

Several open-source tools described in Sec. I A provide cardiac fluid–structure interaction capabilities. SimVascular allows for both 3D–3D and 3D–0D coupling while FEBio and life^x allow for 3D–0D coupling. CircAdapt offers a fully 0D approach to cardiac fluid–structure interaction.

C. Mechanobiology in disease and development

In recent decades, understanding the mechanisms by which the mechanical environment affects the behavior and evolution of cardiovascular structures has become increasingly central to predictive simulations. Early experiments discovered that vascular cells are sensitive to their mechanical environment and seek to maintain homeostatic stress values.^244–247 This sensing is typically attributed to the extracellular matrix of endothelial and smooth muscle cells, which sense fluid shear stresses and intramural stresses, respectively. While the true mechanisms by which this sensing is translated into phenotypical expression remain under investigation, phenomenological models of mechanobiological responses have allowed for increasingly high-fidelity models of both vascular and cardiac growth and remodeling.

Early predictive models took advantage of these biological findings to create kinematic models of vascular and cardiac growth and remodeling that relied on the multiplicative decomposition of growth and elastic components of material deformation.^248–250 Later frameworks of vascular growth and remodeling modeled the turnover of individual constituents in order to provide additional information about the evolving natural configuration of the vessel.²⁵¹ While growth and remodeling frameworks typically work in a continuum description, recent studies have adopted a discrete fiber-based growth and remodeling framework that can provide detail on fiber structure and behavior.²⁵²

Many models of vascular growth and remodeling use reduced-order relationships to calculate the wall shear stress and intramural stress, commonly using the thin-walled cylindrical assumption to calculate intramural stress and using the assumption of Newtonian Poiseuille flow to calculate wall shear stress. Under these assumptions, growth and remodeling trends in response to hypertension, axial stretch, and increased flow have been successfully recapitulated.^253,254 However, there remains a need for clinical prediction of complex, spatially resolved patient-specific growth and remodeling, which, in many cases, results in complex flow fields that do not follow such assumptions. Progress has been made toward fluid–solid-growth frameworks that allow for growth and remodeling theory to be coupled with 3D hemodynamic fields (Fig. 5).^255–257

FIG. 5.

A fluid–solid-growth interaction framework, bridging different physics (fluid dynamics and solid growth) and time scales (milliseconds in fluid dynamics and days in solid growth).

Tissue engineering is an application where modeling vascular growth and constituent turnover has important clinical and design implications. Clinicians have noted the propensity of tissue-engineered vascular grafts to stenose and occlude, but in some cases such stenoses have been observed to spontaneously resolve.^258,259 By expanding vascular growth and remodeling frameworks to include inflammatory response and engineered graft constituent turnover, researchers have been able to capture the evolving in vivo geometries (stenosis and subsequent resolution) exhibited by tissue-engineered vascular grafts in animal models.²⁶⁰

Similarly, cardiac tissue senses its mechanical environment through a mixture of extracellular matrix and internal cellular components, which are connected through focal adhesions and transmembrane integrin proteins. Unique to cardiac tissue are cardiomyocytes which are made up of sarcomeres. While myocytes themselves have a low turnover rate, sarcomeres are able to add and remove unit elements and effectively change the shape and size of a myocyte. Kinematic growth theory remains a popular model of cardiac adaption to mechanical perturbations, with the growth tensor commonly being phenomenologically modeled as a function of the stress and strain tensors. However, such models are limited in their ability to model the prestress of multiple constituents in cardiac tissue.^261–263 Other models of cardiac adaption have been proposed, including those that have modeled the heart as a collection of fibers that can change in length, size, and orientation based on experienced stresses and strain,^264,265 but use of full fluid–structure interaction simulations for calculation of cardiac stresses and strains in service of modeling cardiac adaption remains rare given the expense of calculating FSI in comparatively large domains. A more common approach is utilizing multiscale simulations to reduce the cardiac simulation to a purely structural problem with the fluid modeled as part of a lumped-parameter boundary system.

Mechanobiology also plays a crucial role in embryonic development, with mechanical cues influencing both normal and maladaptive cardiac formation. Experiments using zebrafish first showed the impact of fluid dynamics on cardiac development.²⁶⁶ Embryonic zebrafish are particularly well-suited to the study of cardiac development as they are transparent at the embryonic stage, allowing for longitudinal imaging during critical developmental stages. This has allowed researchers to quantify forces that are not otherwise measurable in the lab due to small scale or limits of imaging. Using 4D light sheet imaging and image registration, researchers have been able to quantify wall shear stresses, and oscillatory shear stress in the developing zebrafish heart help regulate the development of cardiac trabeculation, with high oscillatory shear index values correlated with upregulated endothelial Notch-related mRNA expression.^267,268

Chick embryos have similarly been used to investigate developing vasculature. Methods to create computational fluid dynamics simulations of the embryonic aortic arch were developed utilizing Fluorescent dye injection, micro-CT, and Doppler velocity recordings.²⁶⁹ The effect of occluding various portions of the developing aorta was simulated using CFD, and a wall shear stress-based kinematic vascular growth algorithm was applied to predict the effect of flow changes on vascular growth. While purely wall shear stress-driven growth captured the qualitative trends in geometry, there was still significant variation in the quantitative growth, suggesting that additional biodynamic processes must be accounted for to simulate in vivo growth and development.²⁷⁰

Mechanotransduction also plays a role in the formation of thrombosis. Thrombi are formed through a combination of biochemical and biomechanical reactions that include platelet hemodynamics, platelet–platelet interaction, and platelet–wall interaction. Using patient-specific computational fluid dynamics has allowed researchers to correlate the presence of hemodynamic factors, including reduced wall shear stress and elevated residence time, to the risk of thrombosis,³⁰ but while local hemodynamics can indicate the risk of thrombus formation, true simulation of thrombus creation and proliferation would require modeling free-body platelets and red blood cells in macroscale blood flow and a framework for tracking and applying the microscale forces experienced by activated platelets. Current frameworks have bridged the scales of this micro–macro interaction and produced simulations that couple fluid dynamics to microscopic processes that are modeled through various frameworks, including agent-based and stochastic simulations.^271–274 Work in determining the relative contribution of biochemical vs biomechanical factors remains an ongoing area of research.

Although it is widely accepted that cells seek to promote tissue homeostasis in response to biochemical and biomechanical cues, the ways in which these cues translate into tissue adaption rely on the complex interaction between mechanical sensing, gene expression, and chemical cues. Work has been done toward systems biology approaches that can provide additional details to the cell signaling pathways that yield cardiovascular structural changes.²⁷⁵ Models of cardiac cell-signaling pathways have been coupled to finite element ventricular models to capture the effects of hypertrophy in a framework proposed by Estrada et al.²⁷⁶ In this framework, strain was derived from a finite element simulation of a left ventricle and used as an input into a cell-signaling model previously validated to predict changes in cell area.²⁷⁷ The predicted cell area was then mapped to volumetric growth which was then enforced in the finite element model via a kinematic growth formulation. The experienced strains were then updated and fed into the cell-signaling network to calculate the next growth and remodeling time step. Logic-based models of the coupled biochemical and biomechanical response in arteries have similarly shown qualitative agreement with animal models of hypertension.²⁷⁸ In this framework proposed by Irons et al., intramural stress and wall shear stress derived from a reduced-order simulation of a murine abdominal aorta were used as inputs into a cell-signaling network. Rather than outputting changes in cell area, the cell-signaling network was used to scale the production and degradation gains of specific constituent families, motivated by previous work that validated cell-signaling regulation of vascular composition.²⁷⁹ The vascular wall properties were then updated and the new geometric configuration was found using constrained mixture theory, supplying the stresses fed into the cell-signaling network at the next time step. Although more computationally expensive to scale, this framework has the benefit of tracking vascular composition and considering the relative contribution of constituent families over time. In both frameworks, additional exogenous factors were also used as inputs into the cell-signaling network, creating targets for investigating drug delivery and virtual knockout behavior at the macroscale. Current work in cell-signaling growth and remodeling has not incorporated fully 3D FSI simulations, instead relying on reduced-order simulations that do not consider the effect of a highly resolved fluid field. Coupling cell-signaling pathways to high fidelity FSI solvers is a natural evolution of the field that will allow insight into the behavior of complex, patient-specific cardiovascular geometries over time.

V. FUTURE CHALLENGES

The field of cardiovascular simulations has expanded rapidly in the past few decades, from simplified models of idealized vessels to patient-specific, multiscale simulations. Future developments in the field will continue to expand their diagnostic and predictive power.

One of the earliest applications of cardiovascular simulation was investigating the effect of surgical devices and treatment planning. However, only by moving beyond CFD to consider coupled multi-physics and mechanobiological systems will the true predictive power of simulations be realized. Fluid–solid-growth simulations are already being used to optimize proposed cardiovascular devices, and as the capabilities of such frameworks improve, it will become possible to predict hemodynamic outcomes not only at single timepoints but over days, months, and years. While previous methods of growth and remodeling have relied on reduced-order geometries or simplified boundary conditions, growth and remodeling simulations coupled to fully 3D FSI is on the near horizon. With multiscale boundary conditions, it could then be possible to use age-based scaling laws to account for closed-loop boundary condition responses to changes in local hemodynamics, furthering the predictive power of cardiovascular simulations in long-term patient settings.

As our ability to simulate such complex behavior increases, it will become increasingly necessary to provide high-fidelity parameter inputs to create patient-specific simulations. Though phenomenological drivers of cardiovascular growth remodeling are well established and widely used, most studies rely on assumed or estimated values for homeostatic targets and material properties despite observations that such values can vary significantly across patient populations.²⁸⁰ Increasingly mechanistic models are being developed to describe mechanobiological processes, which have the potential to be driven by patient-specific geometry, genomics, and proteomics. Careful collaboration with vascular and systems biologists will be necessary to identify patient-specific parameters of subcellular interactions. Quantifying cell-signaling parameters has historically been challenging as the complex feedback within a network can obfuscate individual reactions, and current models have relied on aggregating both biomechanical and functional evidence of pathway effects. Advances are being made in quantifying the behavior of specific regulatory pathways including, for example, studies that have quantified endothelial surface density of vascular endothelial growth factor receptors using flow cytometry in both healthy and diseased states^281,282 and identifying phosphorylation pathways that critically regulate ventricular mechanical behavior using gene-targeted mice.²⁸³ These advances demonstrate tools that can be used to further the understanding of multiscale regulation and have direct application to simulation of cardiovascular diseases that are propagated at both the mechanical and cellular level. This also has the potential to expand the ability of CFD to investigate diseases such as diabetes, ischemia, and cancer where angiogenesis and vascular remodeling are intimately linked with disease progression and outcomes. The development of additional tools to identify and validate parameters will be critical for the continued development of multiscale mechanobiological modeling frameworks.

Limitations remain in creating patient-specific models of cardiovascular structures that are beneath the resolution of clinical imaging modalities. While we can utilize idealized trees and scaling laws to make estimates of distal vascular behavior, these typically rely on limited sample sizes or animal models. Validating these techniques against human data is challenging due to the inherent limitations of current imaging technologies. However, progress toward patient-specific modeling of the microvasculature could be made by incorporating experimentally measured perfusion maps to guide the growth of synthetic trees. These algorithms could further be generalized to account for more complex bifurcation patterns in vascular trees. This would allow researchers, for example, to more accurately capture blood dynamics at the capillary level, where vessels with a diameter less than $20\hspace{0.17em}\mu \mathrm{m}$ are normally arranged in a web-like structure featuring loops. In the future, improved microvascular models will be used to estimate perfusion maps (in cases where these are unknown) and to set more physiological boundary conditions of large-scale models.

Given the limited precision of inputs into cardiovascular simulations, quantifying bounds on CFD-driven findings is critical to the clinical translation of simulation studies, particularly in multiscale and long-term cardiovascular simulations. Therefore, developing robust uncertainty quantification tools will be an important priority for the expansion of the translational power of computational cardiovascular medicine. Uncertainty quantification techniques rely on numerous simulations and have been shown to take great advantage of reduced order models in multi-fidelity approaches. Therefore, the development of advanced reduced order models accounting for patient-specific mechanobiology phenomena and gene regulatory networks will lead to the quantification of uncertainty associated with the long-term assessment of treatments and surgical procedures.

The field of cardiovascular modeling has expanded on traditional CFD frameworks by incorporating modeling techniques ranging from mechanobiology to electrophysiology and molecular systems biology. As computational power continues to increase exponentially, the level of detail available for patient-specific hemodynamic models will be limited only by the ability to describe multiscale interactions and to accurately identify parameters governing patient-specific behavior. Even in the absence of absolute certainty in model inputs, increasingly robust uncertainty quantification methods will allow for clinical translation of simulation results by providing well-characterized statistics on model predictions. As in silico experiments become progressively reliable and computationally efficient, we will increasingly have the ability to simulate outcomes on large cohorts of synthetic patients, paving the way toward large virtual clinical trials for FDA approval and creating new pathways for the development of cardiovascular devices and treatment methods.

AUTHOR DECLARATIONS

Conflict of Interest

The authors have no conflicts to disclose.

Author Contributions

Erica L. Schwarz: Writing – original draft (equal); Writing – review & editing (equal). Luca Pegolotti: Writing – original draft (equal); Writing – review & editing (equal). Martin R. Pfaller: Writing – original draft (equal); Writing – review & editing (equal). Alison Marsden: Conceptualization (equal); Funding acquisition (equal); Supervision (equal); Writing – original draft (equal); Writing – review & editing (equal).